Strongly positive semigroups and faithful invariant states (Q788965)

The results on noncommutative ergodic theory are proved in the following setting: M is a \(W^*\)-algebra, \(\{\tau_ t| t>0\}\) a semigroup of strongly positive (i.e. \(\tau_ t(A^*A)\geq \tau_ t(A)^*\tau_ t(A))\) linear maps of M into itself (no continuity assumptions of \(\tau\) as a function of t is required), and \(\omega\) is a faithful \(\tau\)- invariant normal state on M. It is shown that many results known in the case when \(\tau\) is a group of *-automorphisms [\textit{O. Bratteli} and the author, Operator algebras and quantum statistical mechanics, Vol. I (1979; Zbl 0421.46048)] can be extended to this situation. Some results of the paper [\textit{A. Frigerio}, ibid. 63, 269-276 (1978; Zbl 0404.46050)] are also generalized. Among the results obtained in the paper are: i) a description of the set of invariant elements in M; ii) conditions that an invariant state \(\omega\) have a unique decomposition into ergodic states; iii) a criterium of ergodicity of \(\omega\) ; iv) in the case when \(\tau\) is 2-positive, a strong positivity of a semigroup \(| \tau |\) is proved, where \(| \tau |\) is given by \(| \tau_ t|(A)\Omega =| T_ t| A\Omega\) (\(\Omega\) is the cyclic and separating vector associated with \(\omega\) and \(T_ t\) sends \(A\Omega\) into \(\tau_ t(A)\Omega\), \(A\in M)\). It is shown that \(| \tau |\)-ergodicity of \(\omega\) is equivalent to uniform clustering property with respect to \(\tau\) : \(\lim_{t\to \infty}\| \omega '{\mathbb{O}}\tau_ t-\omega \| =0\) for all normal states \(\omega\) '.